We add a short discussion about GS games from [1]}. We refer to their work for further details and pointers.
Fix sequences of sets \(\mathcal {X}_t,\mathcal {Y}_t\)  for \(t\ge 1\) . We consider infinite
games between two players: in each round \(t\ge 1\) , the first player \(\texttt {P}_A\) 
selects an element \(x_t\in \mathcal {X}_t\) , and then player \(\texttt {P}_L\)  selects an
element \(y_t\in \mathcal {Y}_t\) . The rules of the game are determined by
specifying a set \(\mathcal {W} \subseteq \prod _{t\ge 1}(\mathcal {X}_t\times \mathcal {Y}_t)\)  of winning sequences for \(\texttt {P}_L\) . That
is, after an infinite sequence of consecutive plays
\(x_1,y_1,x_2,y_2,\ldots \) , we say that \(\texttt {P}_L\)  wins if
\((x_1,y_1,x_2,y_2,\ldots )\in \mathcal {W}\) ; otherwise, \(\texttt {P}_A\)  is declared the
winner of the game.
